Sum of squares function explained

In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different. It is denoted by .

Definition

The function is defined as

r _{k(n)} = |\{(a _{1,} a _{2,} \ldots, a _{k)} \in Z ^{k} : n = a ^{2} _{1} + a ^{2} _{2} + … + a ^{2 \}|} _{k}

where

| | denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares.

For example,

r _{2(1)} = 4 since 1 = 0 ^{2} + (\pm 1 ) ^{2} = (\pm 1 ) ^{2} + 0 ^{2} where each sum has two sign combinations, and also r _{2(2)} = 4 since 2 = (\pm 1 ) ^{2} + (\pm 1 ) ^{2} with four sign combinations. On the other hand, r _{2(3)} = 0 because there is no way to represent 3 as a sum of two squares.

Formulae

k = 2

See main article: Sum of two squares theorem. The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by

r _{2(n)} = 4 (d _{1(n)-d} _{3(n))}

where is the number of divisors of which are congruent to 1 modulo 4 and is the number of divisors of which are congruent to 3 modulo 4. Using sums, the expression can be written as:

r _{2(n)} = 4 \sum _{d} ( -1 ) ^{(d-1)/2} The prime factorization n = 2 ^{g} p ^{f _{1}} _{1} p ^{f _{2}} _{2} … q ^{h _{1}} _{1} q ^{h _{2}} _{2} … , where p _{i} are the prime factors of the form p _{i} \equiv 1 \pmod 4 , and q _{i} are the prime factors of the form q _{i} \equiv 3 \pmod 4 gives another formular _{2(n)} = 4 (f _{1} +1 )(f _{2+1) …} , if all exponents h _{1,} h _{2,} … are even. If one or more h _{i} are odd, then r _{2(n)} = 0 .

k = 3

See also: Legendre's three-square theorem. Gauss proved that for a squarefree number,

r _{3(n)} = \begin{cases} 24 h( - n), & if n\equiv 3 \pmod{ 8 }, \\ 0 & if n\equiv 7 \pmod{ 8 }, \\ 12 h( -4 n) & otherwise , \end{cases} where denotes the class number of an integer .

There exist extensions of Gauss' formula to arbitrary integer .^{[1] [2]}

k = 4

See main article: Jacobi's four-square theorem. The number of ways to represent as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e.

r _{4(n)=8\sum} _{d\midn, 4\nmidd} d.

Representing, where m is an odd integer, one can express

r _{4(n)} in terms of the divisor function as follows:r _{4(n)} = 8 \sigma( 2 ^{min\{k,1\} }m).

k = 6

The number of ways to represent as the sum of six squares is given by

r _{6(n)} = 4 \sum _{d\mid} d ^{2(} 4 \left(\tfrac{ -4 }{n/d}\right) - \left(\tfrac{ -4 }{d}\right) ) , where \left(\tfrac{ ⋅ }{ ⋅ }\right) is the Kronecker symbol.^[3]

k = 8

Jacobi also found an explicit formula for the case :^[3]

r _{8(n)} = 16 \sum _{d\midn} ( -1 ) ^{n+d} d ^{3.}

Generating function

r _{k(n)} for fixed can be expressed in terms of the Jacobi theta function:^[4] \vartheta( 0 ;q) ^{k} = \vartheta ^{k(q)} _{3} = \sum ^{inf ty} _{n =0} r ^{n,} _{k(n)q}

where

\vartheta( 0 ;q) = \sum ^{inf ty} _{n =-inf ty} q ^{n ^{2}} = 1 + 2 q + 2 q ^{4} + 2 q ^{9} + 2 q ^{16} + … .

Numerical values

The first 30 values for

r _{k(n),} k =1 , ... , 8 are listed in the table below:

n	=	r1(n)	r2(n)	r3(n)	r4(n)	r5(n)	r6(n)	r7(n)	r8(n)
0	style='text-align:center;'	0	1	1	1	1	1	1	1	1
1	style='text-align:center;'	1	2	4	6	8	10	12	14	16
2	style='text-align:center;'	2	0	4	12	24	40	60	84	112
3	style='text-align:center;'	3	0	0	8	32	80	160	280	448
4	style='text-align:center;'	2^2	2	4	6	24	90	252	574	1136
5	style='text-align:center;'	5	0	8	24	48	112	312	840	2016
6	style='text-align:center;'	2×3	0	0	24	96	240	544	1288	3136
7	style='text-align:center;'	7	0	0	0	64	320	960	2368	5504
8	style='text-align:center;'	2^3	0	4	12	24	200	1020	3444	9328
9	style='text-align:center;'	3^2	2	4	30	104	250	876	3542	12112
10	style='text-align:center;'	2×5	0	8	24	144	560	1560	4424	14112
11	style='text-align:center;'	11	0	0	24	96	560	2400	7560	21312
12	style='text-align:center;'	2^2×3	0	0	8	96	400	2080	9240	31808
13	style='text-align:center;'	13	0	8	24	112	560	2040	8456	35168
14	style='text-align:center;'	2×7	0	0	48	192	800	3264	11088	38528
15	style='text-align:center;'	3×5	0	0	0	192	960	4160	16576	56448
16	style='text-align:center;'	2^4	2	4	6	24	730	4092	18494	74864
17	style='text-align:center;'	17	0	8	48	144	480	3480	17808	78624
18	style='text-align:center;'	2×3^2	0	4	36	312	1240	4380	19740	84784
19	style='text-align:center;'	19	0	0	24	160	1520	7200	27720	109760
20	style='text-align:center;'	2^2×5	0	8	24	144	752	6552	34440	143136
21	style='text-align:center;'	3×7	0	0	48	256	1120	4608	29456	154112
22	style='text-align:center;'	2×11	0	0	24	288	1840	8160	31304	149184
23	style='text-align:center;'	23	0	0	0	192	1600	10560	49728	194688
24	style='text-align:center;'	2^3×3	0	0	24	96	1200	8224	52808	261184
25	style='text-align:center;'	5^2	2	12	30	248	1210	7812	43414	252016
26	style='text-align:center;'	2×13	0	8	72	336	2000	10200	52248	246176
27	style='text-align:center;'	3^3	0	0	32	320	2240	13120	68320	327040
28	style='text-align:center;'	2^2×7	0	0	0	192	1600	12480	74048	390784
29	style='text-align:center;'	29	0	8	72	240	1680	10104	68376	390240
30	style='text-align:center;'	2×3×5	0	0	48	576	2720	14144	71120	395136

See also

• Integer partition
• Jacobi's four-square theorem
• Gauss circle problem

Notes and References

1. P. T. Bateman . On the Representation of a Number as the Sum of Three Squares . Trans. Amer. Math. Soc. . 71 . 1951 . 70–101 . 10.1090/S0002-9947-1951-0042438-4 .
2. S. Bhargava . Chandrashekar Adiga . D. D. Somashekara . Three-Square Theorem as an Application of Andrews' Identity . Fibonacci Quart . 1993 . 31 . 2 . 129–133 .
3. Book: Cohen, H. . Henri_Cohen_(number_theorist) . Number Theory Volume I: Tools and Diophantine Equations . 5.4 Consequences of the Hasse–Minkowski Theorem . 2007 . Springer . 978-0-387-49922-2.
4. Book: Milne . Stephen C. . Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions . Springer Science & Business Media . Introduction . 2002 . 1402004915 . 9.